# Isotropic matrix elements of the Boltzmann equation collision integral

## TL;DR

This paper introduces an algorithm for calculating isotropic matrix elements of the Boltzmann collision integral, facilitating isotropic relaxation analysis and serving as a foundation for non-isotropic calculations.

## Contribution

It presents a novel recurrence-based algorithm for computing isotropic matrix elements of the Boltzmann collision integral, useful in kinetic theory.

## Key findings

- Recurrence relations connect isotropic matrix elements and Ω-integrals.
- The algorithm enables sequential calculation of matrix elements.
- Isotropic matrix elements are applicable in physical-chemical kinetics.

## Abstract

We propose an algorithm for calculating matrix elements of the non-linear Boltzmann equation collision integral in isotropic case. These matrix elements are used as starting ones in the recurrence procedure for calculating the matrix elements of the collision integral, which is non-isotropic with respect to velocities, as described in our previous paper. In addition, isotropic matrix elements are of independent interest for the calculation of isotropic relaxation in a number of problems of physical-chemical kinetics. It is shown that the coefficients of the isotropic matrix elements expansion in terms of $ \Omega $ -integrals are connected by recurrence relations that allow us to calculate them sequentially.